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Chapter 5: Q32E (page 263)

TRUE OR FALSE

If||Ax鈫�||猢�||x鈫�||for x鈫�all in Rn, then Amust represent the orthogonal projection onto a subspace Vof Rn.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Determine the definition of an orthogonal matrix.

A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Or it can be said that, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

02

Consider an example

For example,

Suppose A is a square matrix with real elements and of n脳norder and AT is the transpose of A. Then according to the definition, if, AT=A-1is satisfied, then,

A路AT=I

For example, take A=1/200-1/2

By observing that we get,

Axy=12xy猢�xy

But A doesn鈥檛 represent an orthogonal projection onto a subspace V of Rn.

Hence, the given statement is false.

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Most popular questions from this chapter

Consider a linear transformationL from RmtoRn that preserves length. What can you say about the kernel of L? What is the dimension of the image? What can you say about the relationship between n and m? If Ais the matrix of L, What can you say about the columns of A? What isATA? What aboutAAT? Illustrate your answer with an example where m=2and n=3.

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

20.[235046007]

TRUE OR FALSE?If matrix A is orthogonal, then the matrixAT must be orthogonal as well.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

7.[201],[-212],[1800]

Consider a line Lin Rn, spanned by a unit vector

u鈫�=[u1u2...un]

Consider the matrix Aof the orthogonal projection onto L. Describe the ijth entry of A, in terms of the componentsu鈫�iof u鈫�.
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